The proof is an immediate consequence of the fact that the Tychonoff cube \([0,1]^{\mathfrak {c}}\)  endowed with the product topology, is separable (see for instance [1]}). Indeed, let
\(\lbrace x_n: n\in \mathbb {N}\rbrace \) 
be a dense subset of \([0,1]^{\mathfrak {c}}\) . For each \(\gamma <\mathfrak {c}\) , we define the element \(f_{\gamma }\in [0,1]^{\mathbb {N}}\)  by
\(f_{\gamma }(n)=x_n(\gamma ), \quad n\in \mathbb {N}.\) 
The fact that the family \((f_{\gamma })_{\gamma <\mathfrak {c}}\)  satisfies the claim follows by the definition of the product topology.
